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Use the time-harmonic wave convention $\psi e^{-i\omega t}$ and write $\mathbf{z}=(y,z)$, $E(0,\mathbf{z})=E_0(\mathbf{z})$, with a deterministic incident wave envelope. The printed speed ratio is inconsistent with $k=k_{0}n$: the refractive index used below is $n=c_0/c$. Substituting $\psi =e^{i{k}_{0}x}E$ in the Helmholtz equation givesThe paraxial approximation discards ${\partial }_x^{2}E$. The usual weak-fluctuation model also linearizes $n^2-1=2\mu W+O({\mu }^2{W}^2)$, givingDropping the quadratic contrast is an additional weak-fluctuation assumption, not a consequence of small propagation angles. Although the linearized random potential generates attenuation of order ${\mu }^2$, it does not retain every effect of that order in the literal finite-correlation index: the discarded quadratic contrast can also produce a mean wave phase shift. This linearization must precede a Gaussian white noise limit: the square of ideal white noise has no ordinary pointwise meaning.

The split-step Fourier method alternates free-space diffraction, $U_h=e^{hD}$, with a random phase screen,For jointly Gaussian random fields, Gaussian phase averaging gives $\mathbb{E}{e}^{i{\varphi }_h}=e^{-\mathrm{Var}({\varphi }_h)/2}$. More generally, a product of $p$ fields and $q$ conjugate fields picks up $\exp (i{\sum }_a{ϵ}_a{\varphi }_h({\mathbf{z}}_a))$, with signs ${ϵ}_a=+1$ or $-1$. Its screen average is determined entirely by the wave phase covariance matrix. The deterministic diffraction step acts on each coordinate, with opposite signs on factors formed by complex conjugation. This is the basis of the field-moment equations.

There is an important closure qualification. A stationary Gaussian random field need not have independent longitudinal increments. For finite-correlation fluctuations, the unlinearized parabolic wave equation instead givesIn the weak-fluctuation model, the exact first-moment equation isThe last term cannot in general be replaced by a constant times $m$. An exact generic solution of the linearized model iswhere $\mathcal{T}$ denotes time ordering of the propagation operators. For the unlinearized model, add $i{k}_0{\mu }^2{W}^2/2$ inside the propagation generator. The covariance function of the medium is needed to evaluate this expression; One-point Gaussian distributions alone would not even determine the joint wave phase statistics.

For example, omit diffraction and take a longitudinal autocovariance function $R(s)=\mathbb{E}[W(s,\mathbf{z})W(0,\mathbf{z})]$. Direct Gaussian phase averaging givesEven here, unit-variance stationary Gaussian random fields with different $R$ give different answers. In the general problem the diffraction and multiplication operators do not commute, so this scalar attenuation cannot simply be multiplied by $U_x$ without an additional approximation.

The standard closed answer uses the Markov approximation for a random medium, made explicit in part ii. Let the longitudinally integrated covariance kernel beReplace the medium by longitudinal Gaussian white noise with this strength. A screen of thickness $h$ then has $\mathbb{E}[{\varphi }_h({\mathbf{z}}_a){\varphi }_h({\mathbf{z}}_b)]=k_0^2{\mu }^{2}hB({\mathbf{z}}_a-{\mathbf{z}}_b)$ and is independent of the incoming field. For the moment $M=\mathbb{E}{\prod }_a{E}_a^{({ϵ}_a)}$, expanding both steps to order $h$ givesIn particular, the coherent attenuation in a white-noise random medium and its Fresnel propagator solution areEquivalently, write $Wdx=d{\beta }_x$, where ${\beta }_x(\mathbf{z})$ is Brownian motion in $x$ with transverse covariance kernel $\mathbb{E}[d{\beta }_x(\mathbf{z})d{\beta }_x({\mathbf{z}}^{\prime })]=B(\mathbf{z}-{\mathbf{z}}^{\prime })dx$. The Stratonovich integral formulation is $dE=DEdx+i{k}_0\mu E\circ d{\beta }_x$. Its Itô integral form isThe mean of the Itô integral vanishes, independently confirming the attenuation drift. For $x>0$ in two transverse dimensions,Thus a unit plane wave has $m=e^{-\gamma x}$. For a general incident wave envelope, the Fresnel propagator supplies its spreading. The attenuation is redistribution between the coherent and diffuse wave fields, rather than wave absorption: for a field and its conjugate at the same point, the screen contribution in the second-moment equation cancels.

In time reversal acoustics, the array records the incoming signal, reverses each recorded time trace, and re-emits it through the same medium. This is phase conjugation in the frequency domain. For the time-harmonic wave convention $e^{-i\omega t}$, reversing a real time trace replaces its positive-frequency wave amplitude by its complex conjugate. The medium must remain unchanged between recording and re-emission.

Let $f(\mathbf{z})={\Psi }_0(0,\mathbf{z})$ and $G_L(\mathbf{\zeta },\mathbf{z})=G(L,\mathbf{\zeta };0,\mathbf{z})$, with the chosen source and array normalizations incorporated into the Green function. Let $a(\mathbf{\zeta })\ge 0$ be the array's aperture weight, equal to the indicator of its receiving region for an ideal uniform array. The recorded field isBy wave reciprocity, back-propagation has the same Green function with the source and receiver exchanged. Therefore the physically re-emitted, back-propagated wave amplitude isIf $Hf=R$ and $P_A$ multiplies by $a$, then ${\Psi }_B=H^T{P}_A\overline{Hf}$, where $T$ denotes the transpose without conjugation. Taking a final complex conjugate instead defines the adjoint reconstruction $\overline{{\Psi }_B}=H^{\ast }{P}_{A}Hf$. This distinction prevents an erroneous conjugation in the time reversal operator.

For a localized Gaussian beam or acoustic point source in a homogeneous medium, a finite aperture admits a limited range of angles. The focal width is of order $\lambda L/A$ when $A$ denotes the aperture diameter. In a random medium, multiple scattering creates paths with a larger angular spread. Each reversed path retraces its route, and the paths interfere constructively at the source. This can produce a larger effective aperture in time reversal and a narrower focus, even though the unreversed field has a complicated speckle pattern.

This comparison concerns a homogeneous reference medium; a deterministic heterogeneous medium can also provide useful multipath propagation. Suitable scale limits or frequency and spatial averaging can make refocusing self-averaging. Such self-averaging is not automatic for every monochromatic source and every random realization. Wave absorption, changing medium parameters, unresolved paths or poor array coverage can spoil refocusing. With complete capture of the propagating modes and a lossless unitary operator $H$, ideal adjoint reconstruction is already exact in either medium. Random scattering can improve finite-aperture wave focusing through angular diversity. See the regime-dependent analysis in Statistical stability in time reversal.